Problem: Let $f(x)=x^3+3$ and $g(x) = 2x^2 + 2x +1$. What is $g(f(-2))$?
Answer: We note that $f(-2)=(-2)^3+3=-5$, so $g(f(-2))=g(-5)=2\cdot(-5)^2+2\cdot(-5)+1=41.$ Therefore our answer is $\boxed{41}$.